Is conservation of energy a local law in Quantum field theory?

[KleinMoretti]

TL;DR Summary

"The local energy conservation in quantum field theory is ensured by the quantum Noether's theorem for the energy-momentum tensor operator. Thus energy is conserved by the normal unitary evolution of a quantum system."

From Wikipedia,

"The local energy conservation in quantum field theory is ensured by the quantum Noether's theorem for the energy-momentum tensor operator. Thus energy is conserved by the normal unitary evolution of a quantum system."

I know that it is the case in GR that conservation of energy and other conservation laws are relegated to being local only I thought this wasn't the case in quantum field theory.

 

[anuttarasammyak]

TL;DR Summary: "The local energy conservation in quantum field theory is ensured by the quantum Noether's theorem for the energy-momentum tensor operator. Thus energy is conserved by the normal unitary evolution of a quantum system."

I know that it is the case in GR that conservation of energy and other conservation laws are relegated to being local only I thought this wasn't the case in quantum field theory.

I have an opposite idea. In GR covariant derivative divergence of energy is zero but I am not certaion whether it means local conservation of energy. In QM virtual particles or ohter misterious sprocess seems to violate energy conservation but these are pictures in perterbation mehods and I am not sure it does harm energy conservation law.

 

[KleinMoretti]

I have an opposite idea. In GR covariant derivative divergence of energy is zero but I am not certaion whether it means local conservation of energy. In QM virtual particles or ohter misterious substance seem to violate energy conservation but these are pictures in perterbation mehods and I am not sure it does harm energy conservation law.

most people seem to agree that energy is conserved locally in general relativity, most of the debate I have seen is about global conservation some people say that energy is conserved in GR others say its not, as far as I understand its mostly a problem of definitions. This however doesn't answer my question about conservation of energy in QFT

 

[PeterDonis]

I know that it is the case in GR that conservation of energy and other conservation laws are relegated to being local only I thought this wasn't the case in quantum field theory.

You thought what wasn't the case for QFT? That local energy conservation is true? Or that local energy conservation is the only kind that holds?

 

[PeterDonis]

I have an opposite idea. In GR covariant derivative divergence of energy is zero but I am not certaion whether it means local conservation of energy.

It means local conservation of stress-energy, which is the GR version of local conservation of energy.

 

[renormalize]

This however doesn't answer my question about conservation of energy in QFT

Can you you clarify what you mean by "conservation of energy"? Are you asking:

• Can an energy-momentum tensor operator with zero-divergence be built from quantum fields? (The answer is yes.)
• Can this tensor operator be integrated over a spacelike hypersurface to form a four-vector operator of total energy and 3-momentum that is independent of time? (The answer is also yes.)

(see, e.g., Itzykson & Zuber Quantum Field Theory sec. 3-1-2)

 

[PeterDonis]

Can this tensor operator be integrated over a spacelike hypersurface to form a four-vector operator of total energy and 3-momentum that is independent of time? (The answer is also yes.)

More precisely, the answer is yes in flat spacetime, or in a stationary curved spacetime that can be foliated by an appropriate set of spacelike hypersurfaces.

(Note, however, that any such treatment assumes either that we are ignoring any contribution to spacetime curvature sourced by the quantum stress-energy tensor operator, or that we are adopting a semi-classical coupled model in which the spacetime geometry is sourced by something like the expectation value of the quantum stress-energy tensor operator. Both types of models are approximations. We do not have a full theory of quantum gravity so we also do not have a fully self-consistent treatment of energy conservation in QFT including the effects of quantum fields on the spacetime geometry.)

 

[KleinMoretti]

More precisely, the answer is yes in flat spacetime, or in a stationary curved spacetime that can be foliated by an appropriate set of spacelike hypersurfaces.

(Note, however, that any such treatment assumes either that we are ignoring any contribution to spacetime curvature sourced by the quantum stress-energy tensor operator, or that we are adopting a semi-classical coupled model in which the spacetime geometry is sourced by something like the expectation value of the quantum stress-energy tensor operator. Both types of models are approximations. We do not have a full theory of quantum gravity so we also do not have a fully self-consistent treatment of energy conservation in QFT including the effects of quantum fields on the spacetime geometry.)

I thought quantum filed theories were defined in flat Minkowski space and QFT in curved spacetime was more of an intermediate approach to a theory of quantum gravity, but if I understand right conservation of energy is both local and global for a flat spacetimes but you run into challenged in curved spacetimes where a theory of quantum gravity would be needed?

 

[dextercioby]

I thought quantum filed theories were defined in flat Minkowski space and QFT in curved spacetime was more of an intermediate approach to a theory of quantum gravity, but if I understand right conservation of energy is both local and global for a flat spacetimes but you run into challenged in curved spacetimes where a theory of quantum gravity would be needed?

Yes, that's right.

 

[PeterDonis]

I thought quantum filed theories were defined in flat Minkowski space

That is how QFT was originally developed, but QFT is not limited to that.

and QFT in curved spacetime was more of an intermediate approach to a theory of quantum gravity

It is that, but it is also a perfectly good QFT in itself.

if I understand right conservation of energy is both local and global for a flat spacetimes but you run into challenged in curved spacetimes

I specified stationary curved spacetimes; in those curved spacetimes there is a well-defined global notion of energy.

There is also a well-defined global notion of energy (in fact two of them) in asymptotically flat spacetimes.

 

[KleinMoretti]

That is how QFT was originally developed, but QFT is not limited to that.


It is that, but it is also a perfectly good QFT in itself.


I specified stationary curved spacetimes; in those curved spacetimes there is a well-defined global notion of energy.

There is also a well-defined global notion of energy (in fact two of them) in asymptotically flat spacetimes.

In curved spacetimes that are not stationary does conservation of energy run into the same problem as in GR where energy becomes hard to define, also like I said i understand that some people prefer to say that conservation of energy is invalid in general relativity, is this also a valid position for QFT in curved space times that are not stationary?

 

[PeterDonis]

In curved spacetimes that are not stationary does conservation of energy run into the same problem as in GR where energy becomes hard to define

In curved spacetimes that are not stationary or asymptotically flat, yes.

some people prefer to say that conservation of energy is invalid in general relativity

Global conservation of energy, in curved spacetimes that are not stationary or asymptotically flat, yes.

is this also a valid position for QFT in curved space times that are not stationary?

That are not stationary or asymptotically flat, yes.

 

[KleinMoretti]

In curved spacetimes that are not stationary or asymptotically flat, yes.


Global conservation of energy, in curved spacetimes that are not stationary or asymptotically flat, yes.


That are not stationary or asymptotically flat, yes.

In GR there are some ways to salvage GCE in not stationary or asymptomatically flat space times, https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/ , is there something like that in QFT or is Quantum gravity the only solution?

 

[PeterDonis]

In GR there are some ways to salvage GCE in not stationary or asymptomatically flat space times, https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/ ,

Only if you are willing to accept "pseudo-tensors" as physically meaningful. Which many are not since they are frame-dependent, and one of the primary tenets of relativity is supposed to be that frame-dependent quantities are not physically meaningful.

is there something like that in QFT or is Quantum gravity the only solution?

If you are willing to accept a particular "pseudo-tensor" as physically meaningful in a particular curved spacetime classical GR, I don't see why you wouldn't in QFT in the same curved spacetime. But you would still have the same approximation issues that I described before.

 

[Demystifier]

It is that, but it is also a perfectly good QFT in itself.

Well, maybe not perfectly. E.g. the black hole information paradox seems to violate unitarity.

 

[Demystifier]

also like I said i understand that some people prefer to say that conservation of energy is invalid in general relativity, is this also a valid position for QFT in curved space times that are not stationary?

Yes, the latter position is as valid as the former one.

 

[KleinMoretti]

Yes, the latter position is as valid as the former one.

But there is also people who prefer to say that energy is conserved in Gr it just depends on the definition, it's this also a valid position in QFT?
Also when people mention energy non conservation in GR, they use the redshift of photons as they travel through space as an example where energy is "lost". is there any process where energy is "lost" like this in QFT? other than vacuum fluctuations/virtual particles which I know are more of a tool

 

[Demystifier]

But there is also people who prefer to say that energy is conserved in Gr it just depends on the definition, it's this also a valid position in QFT?

Yes.

 

[PeterDonis]

the black hole information paradox seems to violate unitarity.

The only real takeaway from that is that some curved spacetimes might not be compatible with a fully unitary QFT covering the entire spacetime. But that still leaves plenty of others that are. (And indeed, one proposed resolution to the black hole information paradox is that, in a full quantum gravity theory, we will find that there is zero amplitude for the non-compatible curved spacetimes to actually occur, and that what we currently think of as "black holes" are really something more like, say, Bardeen black holes, with no true event horizon and compatible with a fully unitary QFT everywhere.)